Question

A stationary object is released from a point $P$ a distance $3R$ from the centre of the moon which has radius $R$ and mass $M$. Which one of the following expressions gives the speed of the object on hitting the moon?

• A. ${\left(\frac{2GM}{3R}\right)}^{\frac{1}{2}}$
• B. ${\left(\frac{4GM}{3R}\right)}^{\frac{1}{2}}$
• C. ${\left(\frac{2GM}{R}\right)}^{\frac{1}{2}}$
• D. ${\left(\frac{GM}{R}\right)}^{\frac{1}{2}}$

Answer 1

Answer: (B)

${\left(\frac{4GM}{3R}\right)}^{\frac{1}{2}}$

Let the mass of object be $m$ and velocity with which it hits the moon be $v$.

Using conservation of energy at the point P and surface of moon :

$P.E_P+K.E_P=P.E_{surface}+K.E_{surface}$

$\therefore$ $-\frac{GMm}{3R}+0=\frac{-GMm}{R}+\frac{1}{2}m{v}^2$

Or $\frac{1}{2}m{v}^2=\frac{2}{3}\frac{GMm}{R}$

$⟹v=\sqrt{\frac{4GM}{3R}}$